Using Compound Interest to Optimize Investment Spread

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Quantitative Investing Strategies: Compound Interest With Monthly Additions


     The formula for compound interest with monthly additions can be viewed as a finite geometric series.


Mrᵗ Σrᵗ/rⁿ, where the summation is bounded by {n =1, t}


M is the monthly addition

r is the return rate (greater than one)

t is the units of time we'll be using (months, years, etc. - your preference. If t is years, then you have to scale M accordingly).


     Because the equation is a geometric series, we can find its sum through applying the formula


S(n) = a (1 - wⁿ) / (1 - w)


     where a is equal to everything outside our summation, and n stays the same (it can be used as t, because both carry the same value and placement). Note that w is (1/r).


     We end up with the formula


Mrᵗ ( 1 - (1/r)ᵗ ) / ( 1 - (1/r) )


     Adding this to our standard compound interest formula, we create the final formula for compound interest with additions:


Prᵗ + Mrᵗ ( 1 - (1/r)ᵗ ) / ( 1 - (1/r) )


     We can verify this by setting the following example and validating it against the logical sum of the series (taking each unit of time and calculating by hand). Here we enter the data via the formula:


Let P=$10,000


r=1.08


t=2 years


M=$50


I = $11,776.32


Here we enter the data without the formula:


(10,000) rᵗ + 50 ( rᵗ + r ) ==> $11,776.32 ==> Verified!


     Furthermore, we can simplify the compound interest formula down to


Prᵗ + rM( rᵗ-1)/(r-1)


     Now let's analyze whether we should invest all that money now, or over increments of time, via a quantitative/layering value strategy.

Determining when Compound Interest Will Be Higher


     Now we need to determine whether all that money should be invested at the start, or regularly invested every month. Obviously, this guide is not taking risk into account or market fluctuations, but keep in mind that spreading regular payments over months is less risky than putting all your money in at once (where the market could drop the next day and you lose more).


     So we know that our compound interest formula without regular payments is Prᵗ, and that the formula with regular payments is


Prᵗ + rM( rᵗ-1)/(r-1)


     If we set the regular payments formula to be greater than the standard formula, then we can see in which scenarios we are optimizing our investing strategy.


Prᵗ < Prᵗ + rM( rᵗ-1)/(r-1)


     Now, it may be tempting to eliminate the left side, but leaving ourselves with 0 will yield no answer. So let's think about this for a second. The second P is not going to be the same amount of principal: it will be the sum of all the monthly additions subtracted from the initial P, as we are spreading that capital out over the months if we're doing regular payments. Therefore,


Prᵗ < ( P-Mt )rᵗ + rM( rᵗ-1)/(r-1)


     Now it's as simple as solving for M.


Monthly Payments as a Subtraction From Initial Investment


     A quick note: we'll be using the standard compound interest principal capital as P, not the compound interest principal with regular deposits, just as a formatting preference.


Prᵗ <=> ( P-Mt )rᵗ + rM( rᵗ-1)/(r-1)


     Let's test it by guessing that t = 24 months, M=$50, and r is the standard index rate of return per month, 1+.08/12.


P(1.173) < (P-1200)(1.173) + 58.644 (22.2579)


1.173P < 1.173P - 1407.47 + 1305.30


P < P - 102.17


     Therefore, it will always broker larger returns when you invest all of the capital at the start, if you have it (in this scenario). Let's now do a quantitative determination of whether investing all that money at the start brokers larger returns when you have a recurring income stream, but only a certain amount of investing capital (principal) at the start.


     In this example, let's assume that you're willing to set aside $500 each month for investing. On the average developed-nation salary, this won't be hard to do. We're going to be solving for P - your Principal investment - and seeing whether you should save until a certain amount of paychecks, or invest as you go.


Compound Interest With Upcoming Investable Revenue


     We know that your monthly addition to your investments will be around $500 per month (M). However, in this example, P is not totally dependent on M - you have a little bit of money to start, maybe. So let's define P as your initial money to invest ( I₀ ) plus the time ( tₑ )you might wait (we'll see!) to collect a certain amount of your paycheck (this is different than the compounding time).


     So, using our compound interest formula with monthly additions, let's compare if you wait for a couple paychecks before investing, or if you invest what you have at the start and invest a portion of that paycheck when it comes.


Scenario 1: Your monthly paycheck is around the corner, and you want to invest $1000 of each of your paychecks into the market. You decide to wait for after the second payday so that you can invest a lot of money at one time ($2000). You already have $5,000 to invest (that you're not planning on already), but want to hold off until you have that lump sum to invest it all at one time. You're still giving $500 a month to the market.


     Let's do the math for scenario 1 for the end of 2 months.


f( 2-month period ) = rM( rᵗ-1)/(r-1) + lump sum


= (1.0067)(500)(1.0067²-1)/(.0067) + lump sum


= 1010.07 + 7000 = $8010.07


Scenario 2: Even though your monthly paycheck is around the corner, you want to invest early. So you're going to put your $5000 into the stock market now, and invest those $1000 after each respective payday. You're still giving $500 a month to the market.


Scenario 2 after 2 months ( k = payday investments ):


f( 2-month period ) = rM( rᵗ-1)/(r-1) + Prᵗ + krᵗ/r + k


= 1010.07 + 5067.22 + 2006.7


= $8084.00


Scenario 3: You're going to split your money. You're going to invest $4000 at the start, and then $1500 after each payday. You're still giving $500 a month to the market.


Scenario 3 after 2 months ( k = payday investments ):


f( 2-month period ) = rM( rᵗ-1)/(r-1) + Prᵗ + krᵗ/r + k


= 1010.07 + 4053.78 + 3010.05


= $8073.90


     It's this math that favors the relatively short-term investor (2 months). Putting a lot of money into the market early can be a major play - but as time goes on, let's see how each portfolio continues to perform.


Scenario 1 after 6 months:


f( 6-month period ) = rM( rᵗ-1)/(r-1) + Prᵗ + kr( rᵗ⁻¹ -1)/(r-1)


f( 6-month period ) = (1.0067) (500) (6.10) + (7000)rᵗ⁻²


f( 6-month period ) = $10,259.93


Scenario 2 after 6 months:


f( 6-month period ) = rM( rᵗ-1)/(r-1) + Prᵗ + kr( rᵗ⁻¹ -1)/(r-1)


f( 6-month period ) = (1.0067) (500) (6.10) + 5204.40 + (1000) r( rᵗ⁻¹ -1)/(r-1)


f( 6-month period ) = 8274.83 + 5101.40 = $13,376.23


Scenario 3 after 6 months:


f( 6-month period ) = rM( rᵗ-1)/(r-1) + Prᵗ + kr( rᵗ⁻¹ -1)/(r-1)


f( 6-month period ) = (1.0067) (500) (6.10) + 4163.52 + (1500) r( rᵗ⁻¹ -1)/(r-1)


f( 6-month period ) = 3070.435 + 4163.52 + 7652.10 = $14,886.06


     A word of advice: when we're looking at more short-term investments, your cost basis is incredibly important to take into account. When you're dealing with more long-term investments, the cost basis doesn't matter as much, because it's a constant.


     Let's take a look at how the portfolios perform after a decade, now. Definitely a long time-interval jump, but it'll show us what the long-term trends of each of these investing strategies results in. I'll provide the descriptions again below:


Scenario 1: Your monthly paycheck is around the corner, and you want to invest $1000 of each of your paychecks into the market. You decide to wait for after the second payday so that you can invest a lot of money at one time ($2000). You already have $5,000 to invest (that you're not planning on already), but want to hold off until you have that lump sum to invest it all at one time. You're still giving $500 a month to the market. You'll continue this pattern - wait two months, and then invest $2000 from your paychecks - for this decade.


Scenario 1 after 10 years:


f( 1 Decade ) = rM( rᵗ-1)/(r-1) + initial lump sum + kr( rᵗ⁻¹ -1)/(r-1)


f( 1 Decade ) = (1.0067) (500) (183.35) + (7000)rᵗ⁻²  + 1000r (rᵗ⁻¹ -1)/(r-1)


f( 1 Decade ) = 92,418.52 + 15,392.39  + 182,354.84


f( 1 Decade ) = $290,038.89*

     *This number will change slightly due to how you set up the equation. If you staggered investment times, then the rates would be staggered as well, and the time would change. So this is the same value you'd get if you simply invested $1000 of pay each month.


Scenario 2: Even though your monthly paycheck is around the corner, you want to invest early. So you're going to put your $5000 into the stock market now, and invest those $1000 after each respective payday. You're still giving $500 a month to the market.


Scenario 2 after 10 years:


f( 1 Decade ) = rM( rᵗ-1)/(r-1) + Prᵗ + kr( rᵗ⁻¹ -1)/(r-1)


f( 1 Decade ) = (1.0067) (500) (183.35) + 11,142.37 + (1000) r( rᵗ⁻¹ -1)/(r-1)


f( 1 Decade ) =  92,418.52 + 11,142.37 + 182,354.84


f( 1 Decade ) = $285,915.75


Scenario 3: You're going to split your money. You're going to invest $4000 at the start, and then $1500 after each payday. You're still giving $500 a month to the market.


Scenario 3 after 10 years:


f( 1 Decade ) = rM( rᵗ-1)/(r-1) + Prᵗ + 1500r( rᵗ⁻¹ -1)/(r-1)


= 92,418.52 + 4000rᵗ + 1510.05 (181.14)


f( 1 Decade ) = $9,955,326.21



Compound Interest: What Works and What Doesn't


     Why was the third scenario so much more successful over the long run? It had a consistent, relatively high monthly investment rate compared to the other strategies.


     But wait. Let's dial it back and look at each strategy's cost basis.


     Scenario one's cost basis was $183K. That's about 1.5K/month.


     Scenario two's cost basis was $184K.


     Scenario three's cost basis was - get this - $242.5K. Significantly higher. Let's take the returns and quantify how successful each strategy is.


Scenario 1: 290,038.89/183K ==> 58% return


Scenario 2: 285,915.75/184K ==> 55% return


Scenario 3: 9,955,326.21/242.5K ==> 41% return


     This effect is known as large-value radiation loss. Proportionally, as the values increase in size, the returns become comparably smaller, as the divisors are growing as well. Take it as you will. We will be covering an optimization plan for the subscribers shortly.


All Compound Interest Formulas, Compared


     Let's look at all of the formulas we went through today.

The Essentials, Delivered.

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Various compound interests formulas that compare a lump sum vs. regular investment strategy.
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